Orthogonal sequences in non-Archimedean locally convex spaces
Let \(E\) be a Hausdorff locally convex space over a non-Archimedean valued complete field \(K\). A sequence \(x_1,x_2,\dots\) in \(E\) is called `orthogonal' (De Grande-De Kimpe, 1972) if there is a base \({\mathcal P}\) of continuous seminorms such that for all \(n\in\mathbb{N}\), all \(\lambda_1,\dots,\lambda_n\in K\), all \(p\in {\mathcal P}\), \(p(\sum^n_{i=1}\lambda_i x_i)= \max_i p(\lambda_ix_i)\).NEWLINENEWLINENEWLINEThe main result of the paper is the following characterization of compactoids. A bounded subset \(X\) of \(E\) is a compactoid iff each `orthogonal' sequence in \(X\) tends to \(0\). Next, this fact is being to find partial answers to the following questions.NEWLINENEWLINENEWLINE1. If \(E\) is metrizable, does it have an `orthogonal' basic sequence?NEWLINENEWLINENEWLINE2. If \(E\) is Fréchet, of countable type, does it have an `orthogonal' base?NEWLINENEWLINENEWLINENote: Recently, \textit{W. Sliwa} managed to provide full answers [`yes' on question 1, see Indeg. Math., New Ser. 11, No. 3, 463-466 (2000; review below); `no' in question 2, see ``Examples of non-Archimedean nuclear Fréchet spaces method a Schauder basis'', to appear in Indeg. Math., New Ser.].
scientific article; zbMATH DE number 1611979